We present cosmological hydrodynamical simulations including atomic and molecular non-equilibrium chemistry, multi-frequency radiative transfer (0.7-100 eV sampled over 150 frequency bins) and stellar population evolution to investigate the host candidates of the seeds of supermassive black holes coming from direct collapse of gas in primordial haloes (direct-collapse black holes, DCBHs). We consistently address the role played by atomic and molecular cooling, stellar radiation and metal spreading of C, N, O, Ne, Mg, Si, S, Ca, Fe, etc. from primordial sources, as well as their implications for nearby quiescent proto-galaxies under different assumptions for early source emissivity, initial mass function and metal yields. We find that putative DCBH host candidates need powerful primordial stellar generations, since common solar-like stars and hot OB-type stars are neither able to determine the conditions for direct collapse nor capable of building up a dissociating Lyman-Werner background radiation field. Thermal and molecular features of the identified DCBH host candidates in the scenario with very massive primordial stars seem favourable, with illuminating Lyman-Werner intensities featuring values of 1-50 J21. Nevertheless, additional non-linear processes, such as merger events, substructure formation, rotational motions and photo-evaporation, should inhibit pure DCBH formation in 2/3 of the cases. Local turbulence may delay gas direct collapse almost irrespectively from other environmental conditions. The impact of large Lyman-Werner fluxes at distances smaller than 5 kpc is severely limited by metal pollution.
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